Is all quests category only for a brand new character or geared? Or will that be a separate run? Decided to try speedrunning in a game i enjoy and was looking at all the different categories

The category itself is only for new game (but Geared would definitely be a more fun run than a new game) but if you want it as a category just like run it (Missions would be: https://fromdarkhell.github.io/Borderlands-3/#mission-flow (and no need to do Maliwan Takedown since you can run downpatched))